Planetary Gearbox Calculator

Calculate planetary gear train ratios using the Willis equation. Multi-stage epicyclic analysis with speed, torque, load sharing, and efficiency.

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▸Formula · assumptions · example(ISO 6336 (Epicyclic) · AGMA 6123-B06 · DIN 3960)

Formula

i = 1 + \frac{z_{\text{ring}}}{z_{\text{sun}}} \quad (\text{Carrier output, ring fixed})

Willis fundamental kinematic equation for epicyclic gear trains, planet load sharing, torque multiplication, and stage efficiency.

Assumptions

  • · Planet gear meshing assembly condition satisfied: (z_sun + z_ring) / N_planets = Integer
  • · Floating sun or carrier geometry ensures uniform load sharing among planet gears
  • · Single stage mechanical efficiency estimated at η ≈ 97-98%

Worked example

Worked example — 3-Planet Reducer (z_sun = 18, z_planet = 27, z_ring = 72)

Gear ratio i = 1 + 72/18 = 5.0:1. Input torque of 20 N·m at 3000 rpm yields 97.0 N·m output torque at 600 rpm (η = 97%).